Moduli Spaces: New Directions

代数几何研究各种形状：由多项式方程系统定义的形状。参数的多项式约束在数学(例如，数论中的丢番图方程，如著名的费马方程)和从物理学到工程和代数统计的更广泛的应用中都是非常常见的。在物理学中，代数变量被用来描述各种配置空间(甚至宇宙的形状！)。多项式固有的刚性，特别是与拓扑学和微分几何中允许的更灵活的变换相比，使得应用来自代数(环和模的研究)和偶数理论的广泛技术成为可能，例如模素数的约化。代数几何中最基本的问题之一是学习如何对代数簇的几何性质进行分类和区分。除了离散参数，大多数是拓扑学性质的，代数簇也有连续的参数，称为“模”。例如，黎曼曲面(如球面或环面)按孔数(亏格)进行拓扑分类。Riemann已经知道，它们也有3G-3连续复参数(g是亏格)。值得注意的是，这些参数可以解释为“模空间”上的坐标，“模空间”是编码给定拓扑形状的所有可能几何结构的主空间。PI是低维变体(曲线和曲面)的模空间方面的专家。该项目将侧重于开发研究它们的新技术和方法。第一种方法是基于凸几何，它研究由线性不等(如多边形和多面体)定义的形状。牛顿的一个想法是捕捉一个独特的函数系统的渐近行为，以构造一个曲线几何的多面体近似。另一种方法是线性化。派生范畴将几何线性化，就像函数的切线近似图一样。PI的目的是理解几个模空间的派生范畴。最后，通过对模素数进行约化，可以利用数论来研究模空间。为此，模空间必须配备额外的数据，以同时参数化复杂和有限的几何图形。除了推进代数几何，拟议的研究计划还通过招聘、培训和监督研究生和本科生，并为他们的研究、教学和专业发展提供资源，为发展一支多样化的、具有全球竞争力的STEM劳动力队伍做出贡献。特别是，该项目包括三个专门为REU(本科生的研究经验)量身定做的问题以及几个博士论文问题。PI还将继续发展研究生和本科课程，包括旨在向更广泛的公众介绍数学和科学的课程。PI将致力于代数几何中的几个基本问题，重点是局部系统、曲线和曲面的紧致模空间。PI以前发展了代数环面中子簇的经典“交换”热带化为模空间的应用。最近，PI和他的研究生Vogianyu将经典热带几何推广到“非对易”环境，PI建议使用这种推广来构造穿孔Riemann球面上局部系统的模空间的几何意义紧致。与Castravet合作，PI打算验证Orlov和Kuznetsov关于稳定有理曲线(及相关空间)的模空间的派生范畴的一个猜想，并应用这一描述找出它的非对易变形。PI还打算利用Halpern-Leistner发展的窗口理论来解决关于Toric和log Fano簇的派生范畴的一些公开问题。在与Freixas I Montplet的合作中，PI建议系统地发展曲线的模空间的阿拉克洛夫几何，并将其应用于数论中的问题。PI与Urzua合作提出了几个关于曲面的模的新方案，包括构造具有QHD奇点的曲面的模空间(允许有理同调圆盘Milnor纤维的光滑化)以及循环商奇点在回答Kollar问题的方向上变形的全空间族的反转。此外，还有三个项目是为Reus量身定做的：环面和椭圆直纹曲面上的Seshadri常数，以及超平面排列的补集的对数正则紧凑化方程。

Algebraic geometry studies varieties: shapes defined by systems of polynomial equations. Polynomial constraints on parameters are quite common both in mathematics (for example, Diophantine equations in number theory, such as the famous Fermat equation) and in broader applications ranging from physics, where algebraic varieties are used to describe various configuration spaces (and even the shape of the universe!), to engineering and algebraic statistics. The inherent rigidity of polynomials, especially compared to more flexible transformations allowed in topology and differential geometry, makes possible to apply a wide range of techniques coming from algebra (the study of rings and modules) and even number theory, for example reduction modulo prime numbers. One of the most basic problems in algebraic geometry is to learn how to classify and distinguish geometric properties of algebraic varieties. In addition to discrete parameters, mostly of topological nature, algebraic varieties also have continuous parameters called "moduli". For example, Riemann surfaces (such as a sphere or a torus) are classified topologically by the number of holes (the genus). It was known already to Riemann that they also have 3g-3 continuous complex parameters (g is the genus). Quite remarkably, these parameters can be interpreted as coordinates on the "moduli space": the master space that encodes all possible geometric structures of a given topological shape. The PI is an expert on moduli spaces of low-dimensional varieties (curves and surfaces). The project will focus on developing new techniques and approaches to their study. The first approach is based on convex geometry, which studies shapes defined by linear inequalities (such as polygons and polyhedra). An idea that goes back to Newton is to capture asymptotic behavior of a distinguished system of functions to construct a polyhedral approximation of a curved geometry. Another approach is linearization. Derived categories linearize geometries just like tangent lines approximate graphs of functions. The PI aims to understand derived categories of several moduli spaces. Finally, moduli spaces can be studied using number theory by reducing them modulo prime numbers. To this end, moduli spaces have to be equipped with extra data to parametrize complex and finite geometries simultaneously. In addition to advancing algebraic geometry, the proposed research program contributes to the development of a diverse, globally competitive STEM workforce through recruiting, training, and supervising of graduate and undergraduate students and providing resources for their research, teaching, and professional development. In particular, the project includes three problems specifically tailored for REU (Research Experience for Undergraduates) as well as several PhD thesis problems. The PI will also continue to develop graduate and undergraduate courses, including courses designed to introduce a broader public to mathematics and science.The PI will work on several fundamental problems in algebraic geometry with a focus on compact moduli spaces of local systems, curves and surfaces. The PI has previously developed applications of classical "commutative" tropicalization of subvarieties in algebraic tori to moduli spaces. Recently, classical tropical geometry was extended by the PI and his graduate student Vogiannou to a "non-commutative" setting and the PI proposes to use this generalization to construct a geometrically meaningful compactification of the moduli space of local systems on a punctured Riemann sphere. In collaboration with Castravet, the PI intends to verify a conjecture of Orlov and Kuznetsov on derived category of the moduli space of stable rational curves (and related spaces) and to apply this description to find its non-commutative deformations. The PI also intends to use the theory of windows as developed by Halpern-Leistner to solve some open problems about derived categories of toric and log Fano varieties. In collaboration with Freixas i Montplet, the PI proposes to systematically develop Arakelov geometry of the moduli space of curves and to apply it to questions in number theory. The PI proposes several new projects on moduli of surfaces in collaboration with Urzua, including constructions of moduli spaces of surfaces with QHD singularities (admitting smoothings with rational homology disc Milnor fiber) and families of antiflips of total spaces of deformations of cyclic quotient singularities in the direction of answering a question of Kollar. In addition there are three projects tailored for REUs: Seshadri constants on toric and elliptic ruled surfaces, and equations of log canonical compactifications of complements of hyperplane arrangements.